Approximating signals in the abstract

Abstract

An analogue of the classical approximate sampling theorem is proved for the abstract analogue of a signal, i.e. a function on a locally compact abelian group that is continuous, square-integrable with an integrable Fourier–Plancherel transform. An additional hypothesis that the samples of the function are square-summable is needed and is discussed. This hypothesis is not very restrictive as in a sense it ‘almost always’ holds. Two asymptotic formulae are also obtained under some further conditions on the group.

Keywords:

• locally compact abelian groups
• abstract harmonic analysis
• sampling theory
• an approximate Kluvánek theorem

AMS Subject Classifications::

• 94A20
• 43A25
• 94A05

Acknowledgements

The Institute of Biomathematics and Biometry and the organisers of the ‘Approximation Theory and Signal Analysis’ workshop at Lindau to celebrate Paul Butzer's 80th year are to be congratulated for a most enjoyable and stimulating occasion, enhanced by the spectacularly beautiful setting of Lake Constance. Paul has been a friend and colleague for many years and so I am particularly grateful to the Institute for its hospitality and for the financial support which enabled me to participate. The friendly and willing team of helpers deserve special thanks for dealing so efficiently and expertly with endless requests for all sorts of information and assistance. I also thank Rowland Higgins for many helpful conversations.